Consider the following system of equations:

[tex]\begin{aligned}
2 x-y & =12 \
-3 x-5 y & =-5
\end{aligned}[/tex]

The steps for solving the given system of equations are shown below:

[tex]\begin{array}{rlrl}
& \text { Step 1: } & -5(2 x-y) & =-5(12) \
& -3 x-5 y & =-5 \
& \text { Step 2: } & -10 x+5 y & =-60 \
& -3 x-5 y & =-5 \
& \text { Step 3: } & -13 x & =-65 \
& \text { Step 4: } & x & =5 \
& \text { Step 5: } & 2(5)-y & =12 \
& \text { Step 6: } & y & =-2
\end{array}[/tex]

Solution: (5,-2)

Select the correct statement about step 3.

Step 1: Multiply the first equation by -5 to prepare for elimination.
Step 2: Add the modified equations to eliminate y .
Step 3: Simplify the resulting equation to solve for x .
Step 4: Conclude that Step 3 is correct.

Step 3 is correct. ​
Explanation

Problem Analysis We are given a system of equations and a step-by-step solution. Our task is to verify the correctness of Step 3. The system of equations is:

2 x − y − 3 x − 5 y ​ = 12 = − 5 ​
The provided steps are:

Step 1: Multiply the first equation by -5.
Step 2: State the modified system of equations.
Step 3: State the result of adding the two equations from Step 2.
Step 4: Solve for x .
Step 5: Substitute the value of x back into the original equation.
Step 6: Solve for y .

We need to check if Step 3 is correct.

Verifying Step 2 Step 1 involves multiplying the first equation by -5:

− 5 ( 2 x − y ) = − 5 ( 12 )
− 10 x + 5 y = − 60
The second equation remains unchanged:
− 3 x − 5 y = − 5
So, Step 2 correctly states the modified system of equations.

Checking Step 3 To get to Step 3, we add the two equations from Step 2:

( − 10 x + 5 y ) + ( − 3 x − 5 y ) = − 60 + ( − 5 )
Combining like terms, we get:
( − 10 x − 3 x ) + ( 5 y − 5 y ) = − 65
− 13 x + 0 y = − 65
− 13 x = − 65
This matches Step 3. Therefore, Step 3 is correct.

Conclusion The question asks us to select the correct statement about step 3. Since we have verified that step 3 is correct by adding the equations in step 2, the correct statement would be that step 3 is correct.

Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For example, a company might use a system of equations to model its costs and revenues, and then solve the system to find the production level at which costs equal revenues. This helps the company make informed decisions about pricing, production, and resource allocation. Another example is in electrical engineering, where systems of equations are used to analyze circuits and determine the currents and voltages at different points in the circuit.

Answered by GinnyAnswer | 2025-07-03

Step 3 correctly states that -13x = -65, derived from the addition of the modified equations. This confirms that the operations performed in the previous steps were accurate, leading us to the correct value of x. Hence, Step 3 is correct.
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Answered by Anonymous | 2025-07-04